Changes

A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors.  Consider a universe <math>[\mathcal{X}].</math>  Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math>    In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math>

A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors.  Consider a universe <math>[\mathcal{X}].</math>  Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math>    In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math>

We may analyze ''X''² = {‹''u'', ''v''› : ''u'', ''v'' &isin; ''X''} into two parts, specifically, the pairs that lie on and off the diagonal:

We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:

: ''X''² = {‹''u'', ''v''› : ''u'' = ''v''} &cup; {‹''u'', ''v''› : ''u'' &ne; ''v''}

: <math>X^2 = \{ (u, v) : u = v \}\ \cup\ \{ (u, v) : u \ne v \}</math>

In symbolic terms, this partition may be expressed as:

In symbolic terms, this partition may be expressed as: